Some Examples of Automorphism in a Limited Group

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International Journal on Integrated Education

Some Examples of Automorphism in a Limited Group X. Yo. Najmiddinova DSc, Associate Professor of Namangan State University Uzbekist X. Yu. Toxirjonova 2nd year master of Namangan State University Uzbekistan Abstract: Some examples of automorphism in a group with limited liability are considered in the article, a brief analysis is done.

Keywords: Automorphism, element, usually denoted, numbers package.

--------------------------------------------------------------------------------------------------------------------We will first describe the concepts related to this topic.

If X and U are optional divided into sets, each x X If an element with a definite value y U is matched for an element, then X is given a reflection (and is usually denoted as : X U ) and the element u is called the inverse of the element x , and x is the original of the element u , and this is the reflection. is written as u = (x) .

Definition. : X U reflection X set to U set in zaro one precious reflection is called if each one in U element X is the only one in fact owner if any.

Definition. Optional _ from the elements in a structured G set detected  binary operation for q in the house conditions if _ G,  couple or G group is called: 1) associative : a, b , c G for ( a  in ) c = a  ( in  c ) 2) So it is in G e element is present then for it e  a = a  e = a equality optional a G for the element if; such e element G grouping unit (neutral) element is called.

3)  for a G element G has such an element a -1 then for him a  a -1 = a -1  a = e equality if.

Such element a element is called the inverse element.

For example. All in all numbers package Z is the usual defined in it form a group in relation to the tumor action. Adding integers has an associative property It is known from the laws of arithmetic q, that the number zero is the function of the neutral element in this set does ( n Z for 0 + n = n + 0 = n ). In addition, for each n element, there will be an opposite element - n ( n + (- n ) = (- n ) + n = 0). this group all of numbers to add relatively group and is denoted by ( Z , +) .

The basic set is limited from the elements formed found groups limited groups is called in the group elements sony grouping procedure is called An infinite number of elements A group that includes itself is called an infinite group .

Definition. G 1 and G 2 are given in 2 groups get and G 1 group elements to G 2 group elements hand washer so reciprocal a q valuable  reflection let G 1 basic binary application G2 basic binary instead let, that is, if G is in group 1( a ) = a , ( v ) = v , ( c ) = c and a * v = c if it is h ahead of G 2 in a group a ° v = c get _ In that case   G 1 G 2 hand washer isomorphism isomorphism is established groups reciprocal isomorphic groups is called.

Reciprocal a q valuable reflection - being an isomorphism condition again q at home write you can:  a, v For G 1 ( a v ) = ( a ) ° ( v ), this here a v the product is obtained at G 1 , ( a ) Copyright (c) 2022 Author (s). This is an open-access article distributed under the terms of Creative Commons Attribution License (CC BY).To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/

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International Journal on Integrated Education ° (

c ) multiplication is obtained in G 2 .

Definition. G group, g - his elements get _ G groups self to himself conductive g reflection option q at home: g ( h ) = ghg -1 (where h grouping optional elements).

This is it from the reflection g element structured G group internal automorphism is called.

Grouping internal automorphism group self to himself conductive isomorphism be ordinary _ Indeed, ghg -1 = h 1 is only and only if h = g -1 h 1 g . Therefore, as a result of the reflection of g , -1 is a the origin of the element h 1 exists and becomes unique. It follows that the reflection g ( h ) = ghg self-conducting reciprocal one-valued reflection of the group. In addition, g ( h 1 h 2 ) = g ( h 1 h 2 ) g -1 = gh 1 ( g -1 g ) h 2 g -1 = ( gh 1 g -1 ) ( gh 2 g -1 ) = = g ( x 1 ) g ( x 2 ).

Shuning uchun - isomorphism of white.

These concepts are important concepts of abstract algebra. Delivering them to students poses unique challenges. With this in mind, the use of regular polygon substitution groups in the plane in explaining internal automorphisms in finite groups gives students some convenience.

For example. Regular triangle symmetry group Let's look at q ( Figure 1 ). Of a triangle with ends A, B, C. if we define it, it is h ahead this grouping elements three h of the letters A, B, C. place exchange a q value. For example , from end A to VS lowered height relative symmetry  ABC   in view is written .  ACB

1 - picture Triangle symmetry in the group two elements multiply for first place exchange from the second then do need will be . This is it h in front , triangle symmetry group and three h of the letters A, B , C. place exchanges group in the middle installed isomorphism has we will This isomorphism is one q valuable that it is not clear emphasize : it is a triangle exactly q which end with A , which end q with V and C. to determine. Triangle tips designation self h am A, V, S h of the letters place exchange as possible . It 's a corner symmetry in the group h each element of the letters A, B, S h place exchange in view new to determine has will be .

Triangle symmetry in the group all possible divided inside automorphisms as a result to heights relative symmetries k aysi to the elements Let 's take a look at the issue.

Symmetries with respect to heights:  ABC  ABC  ABC c , d   , f   ,  ACB  CBA   BAC 

1) a ( c ) = aca -1 = (BCA) (ACB) (CAB) = (CBA) (CAB) = (BAC) = f 2) b ( c ) = bcb - 1 = (CAB) (ACB) (BCA) = (BAC) (BCA) = (CBA) = d 3) c (c) = c c c -1 = a c -1 = a c = c 4) d (c) = d c d -1 = b d = f Copyright (c) 2022 Author (s). This is an open-access article distributed under the terms of Creative Commons Attribution License (CC BY).To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/

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5) f (c) = f c f -1 = a f = d 6) a (d) = a d a -1 = c b = f 7) b (d) = b d b -1 = f a = c 8) c (d) = c d c -1 = a c = f 9) d (d) = d d d -1 = b d = d 10) f (с) = f d f -1 = c 11) a (f) = a f a -1 = c 12) b (f) = b f b -1 = d 13) c (f) = c f c -1 = d 14) d (f) = d f d -1 = c 15) f ( f ) = fff -1 = f Hence, as a result of all possible internal automorphisms in the group of triangular symmetries, the symmetries with respect to the heights shift to all the height symmetries. let us determine which elements the conversion of a triangle to 120 possible internal automorphisms in the group of triangle symmetry .

0

results in as a result of

 ABC a ,  BCA 

1) a (a) = a a a -1 = b b = a 2) b (a) = b a b -1 = e a = a 3) c (a) = c a c -1 = d c = b 4) d (a) = d a d -1 = b 5) f ( a ) = faf -1 = b 6) c ( a ) = eae -1 = a conversion of a triangle to 120 0 as a result of possible internal automorphisms in the group of triangle symmetry Answer : a - 120 0 and b - 240 0 to convert passes .

The same problem can be solved for other regular polygons. For example, when this problem is considered in the group of tetrahedral symmetry, if all its elements are divided into the following classes: 1) e ; 2) all rotations around the heights other than e ; 3) 180 ° rotation around the arrow passing through the opposite edge ; 4) symmetry with respect to the plane passing through any edge and the edge opposite it; 5) all rotations resulting from the cyclic displacement of the ends  ABCD (for example   ), in which case the two elements in the group of tetrahedral symmetry  BCDA  can pass to each other as a result of internal automorphism, and only when they can be summed up in one class.

It should be noted that in an arbitrary internal automorphism of a group (such as an arbitrary isomorphism), each group of its parts, in general, passes into a completely different group (for example, the symmetry of a triangle relative to the height of one side passes to the symmetry of another height). However, some “special symmetric” subgroups remain in place as a result of internal automorphism (for example, in the triangular symmetry subgroup of a triangle symmetry group).

Copyright (c) 2022 Author (s). This is an open-access article distributed under the terms of Creative Commons Attribution License (CC BY).To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/

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In conclusion, given the specific complexity of the concepts of abstract algebra, their delivery to students, the formation of imagination and skills related to the practical application of these concepts requires a specific approach, the choice of practical problems is important.

List of used literature 1. Р.Н.Назаров ва бошк. Алгебра ва сонлар назарияси / Тошкент, “Ўқитувчи” нашриѐти, 1992 йил.

2. Г.Курош Олий алгебра курси / Тошкент, “Ўқитувчи” нашриѐти, 1992 йил.

3. Э.Фрид. Введение в абстрактную алгебру / Москва, “Мир”, 1979 г.

4. В.Б.Алексеев. Теорема Абеля в задачах и решениях / Москва, “Наука”, 1976 г.

5. Akhmedov, B. A. (2022). Principles of Developing the Professional Competence of Future Teachers on the Basis of a Cluster Approach. Galaxy International Interdisciplinary Research Journal, 10(6), 760-770.

Copyright (c) 2022 Author (s). This is an open-access article distributed under the terms of Creative Commons Attribution License (CC BY).To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/

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